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Explaining MPH And Other Things
Brian D. Rude
Here's a link to a video that I find interesting, and relevant to the teaching of math.
The woman in this video is trying very hard to figure out something very simple, and failing comically. Does that mean she is of low intelligence? Or is there some explanation for her failure that is consistent with normal intelligence?
I have transcribed the conversation as follows. Transcribing from a clip is rather time consuming and tedious, and some words are hard to hear on the clip, so I may have a few mistakes. The woman's name is given in the clip, Chelsea, and denoted as "C" in the transcript below. The man's name is not given so I have simply used "G", for "guy" in the transcript. The numbers before some of the turns in the conversation refer to the time on the video clip.
0:00 G: I just proposed a math question to my beautiful wife, Chelsea, and the question, very difficult to calculate, is, if you are were traveling 80 mph, how long does it take you to go 80 miles? Okay, what do you think, Chel? Let's go through the . . . process, what do you think?
0:29 C: Well, . . . I run the mile in about nine minutes.
G: Wait, what about the tire turning thing you were talking about? You think that affects it?
0:39 C: Well I would guess maybe .. probably . . about, . . . it turns about 400 times in a mile
G: That's just calcu. . . like guessing calculations?
C: Yeah, I don't know how you would work that out.
G: It would be tough.
C: Yeah, so . . . I don't know, . . because if I run a mile in like . . nine minutes, . . then . . . . I mean by that . . . . seven minutes . . . . I'm really
And that takes me a mile, and we're going eighty miles
I'm running at about . . probably ten miles per hour, if that.
C: And that's pretty fast for a human being, I think
But I mean . . .it's gotta take . . . fifty eight minutes or something
1:27 G: fifty eight minutes, that's pretty close
C: to go eighty miles
G: Do you want to know what the answer is?
G: Think about eighty miles per hour, . . . so how long does it take me to go eighty miles since I'm driving eighty miles per hour.
1:45 C: You're driving fast . . . .You are driving faster than a minute a mile
G: Yeah, totally.
1:53 C: So I would whack eighty in half, and that's forty. Cause I mean you're probably going two minutes a mile
2:03 G: Just whack it in half.
C: Well, it depends on. I mean some tires turn faster than other
G: Yeah, it definitely affects it.
C: Cause I mean I think a truck probably is lower to. .
But I think It also matters if your car's a stick or automatic.
2:24 G: Okay, Chel,
G: Think about the term, miles per hour
2:28 C: mph
G: So if I say I'm driving eighty miles, that means I going to go eighty miles in an hour.
2:37 C: No you are not, because you are driving faster than a mile a minute. I can run one mile in seven minutes, and that's a ten. Okay? So I mean . . . and so I'm . . that's booking it too.
G: Okay if I'm going sixty miles per hour.
C Then you're probably, it'll take about 30 minutes
I would just cut it right in half.
3:05 G: I'm trying to explain the answer, but . . .
3:09 C: Well, you're not making sense. because, I make sense. You do not make sense. My first tire may be a little thicker and a little flat so they may not move as fast as . . like a police officers tires that are full . . . but . . . . . . I just . . .
3:25 : Okay, Chel, the answer
C: My tires are turning about . . maybe, I'd say about 400 times per mile.
G: Okay I'm gonna tell you the answer. Are you ready for the answer
C: like a thousand
G: You ready for the answer? The answer to the questions, how far, no I mean,
3:37 C: You don't know the answer!
G: I do know the answer
C: You can guess the meaning, just like I guess the meaning.
C: I guess the meaning . . . I'm using math, science . . . I'm using my own terms
G: Okay, Chel, the answer to the question how far . . .no how . . . how long does it take me to travel eighty miles if I'm driving eighty miles per hour . . . . . the answer is one hour. Your guess was close
C: Okay so say you drive 60 miles per hour at sixty miles . . you want to go sixty miles at sixty miles per hour. Is that an hour?
C: What about forty miles at forty miles per hour
G: Then you will go forty miles in one hour
C: no you don't, you cut that in half . . You're going more than a mile a minute.
C: Okay let's time, . . right now . . . let's time and see.
First we should ask if perhaps the whole conversation is a put on. Perhaps for some reason unknown to us she is acting. Perhaps she is playing the "dumb blonde" just as a joke. I cannot rule this out entirely, but if she is acting she is doing so very convincingly. By all appearances it is a genuine conversation.
Perhaps one bit of evidence that this is not a genuine and candid conversation is the existence of the clip at all. They must have a camera mounted on the dash of their car. How is the camera controlled? Do they always have a camera on the dash of the car ready to record? Maybe the whole thing is scripted and not at all genuine.
But if the conversation is candid and genuine then we might ask how this can be. How can she really not know how long it takes to travel eighty miles if you're traveling at eighty miles per hour? Several hypotheses might be investigated. Perhaps she is of low intelligence and really cannot understand what miles per hour means. Or perhaps something is missing in her education. Or perhaps there is something peculiar about how her brain works. Or perhaps she is just stringing her husband along, not following a script, but not really being honest in her thoughts either.
There is something rather similar to this that I have run across at times, and is the main reason I think this clip is worth thinking about. The question, "How many thirds make a whole" has something in common with the question "How long does it take you to go eighty miles if you are traveling eighty miles per hour". In both cases the answer should be obvious. But I have evidence from experience that "how many thirds make a whole" is question that is not obvious to every one.
When should it be obvious to a child how many thirds make a whole, or how many sevenths make a whole? And, similarly, when should it be obvious to a child how long it takes to go ten miles at ten miles per hour? One might argue, in either of these cases, it should be obvious immediately, because the answer is a logical consequence of the definitions of the terms. One might argue that the word "half" is not a math term. Rather it is simple word of common vocabulary. It should be learned years before a child starts to seriously study fractions. If you know what "half" means, then of course you know that two halves make up a whole. If you don't know that two halves make a whole, then you don't know what a half is. And similarly, if you know what "third" means, then it is obvious that three thirds make up a whole. Again, if you don't know that three thirds make up a whole, then you don't know what a third is. But can this argument be extended indefinitely?
If a child reads in a story something like "each of the robbers expected to receive a fifteenth share of the loot", is it obvious to that child that fifteen such shares would make up the whole of the loot? And is that the same as it being obvious how many fifteenths are in a whole? And is that the same as it being obvious how many fifteenths there are in one? All these would seem to be logically equivalent to the problem of one divided by one fifteenth. But is the answer to that obvious to the child who fully understands the words "half" and "third"? A lot of things that can be demonstrated to be logically equivalent, do not necessarily seem so at first thought.
I do not remember, in my elementary school days, ever doing arithmetic problems involving miles per hour. I am very aware that I could have, but just forgot. The first problems involving miles per hour that I remember were in first year algebra in the ninth grade, distance problems of the type "train A leaves Chicago traveling south at fifty miles per hour, and train B . . . . ." I don't remember having any trouble with these problems at that time. But as a math teacher I am painfully aware that others do. As a math teacher I have struggled with students with these problems. They have struggled to learn to do them while I have struggled to figure out why in they world it is so hard. I have come to a few conclusions, which I have discussed in my article, "Teaching Written Problems In Algebra" which is on my website at http://www.brianrude.com/writpb.htm. The general problem of learning to do written problems in algebra, of course, is broader than understanding problems involving miles per hour.
I will return to the related problem I mentioned, understanding how many thirds make a whole, or how many sevenths make a whole. How do you help a student in a college freshman math class for whom it is not all obvious how many thirds make up a whole? I don't know. But I have been in that position. At least I think I have been in that position. But it is not a clear cut situation. I cannot remember specific students like that, but a few times I thought I was in that situation. This would happen in the context of a student coming to my office to ask for help with the algebra I was teaching. Something would lead me to ask "how many thirds are there in a whole" as a way of getting to something about the algebra. But in that context the question could be easily misinterpreted. A quizzical look on the student's face is not conclusive evidence that the student really doesn't realize that three thirds make a whole by definition of a third. The quizzical look on the student's face may reflect more the student's wondering and where I am going with that question. So I am not sure whether it is true or not that some college students in my algebra classes don't know how many thirds make up a whole.
I will try to describe a little better what I mean by "thinking definitionally". When we hear the words "red box" the normal interpretation is that we know we are talking about a box, and that it is red. We are simply applying the meanings of the words; interpreting them in accordance with customary rules of grammar. But it may not be that simple. Consider this scenario: In a factory at some point in time red boxes are used to throw away scraps of lumber, and blue boxes are used to throw away scraps of plastic. The boxes are spray painted either red or blue. The color coding helps in recycling scrap material. However over a period of time the red boxes become different than the blue boxes. Let's say experience leads to lumber scrap boxes being large and heavily built, capable of holding big scraps of wood, short lengths of two by fours, let us say. But the plastic scrap boxes are small and light, because most plastic scraps are no bigger than a tea cup. Let us further imagine that after a time the boxes are no longer spray painted either red or blue. Their size, construction, and perhaps their locations are obvious clues to anyone with scraps to toss in them. But as a final link in this scenario let us suppose the terms "red box" and "blue box" remain stuck in the language in the factory long after anyone spray painted the boxes to color code them. A worker in this factory, receiving instructions to "throw this in the red box" will not think definitionally about the term "red box". In this context the term "red box" is interpreted as one term, a single noun that happens to be a two word term, a scrap lumber box. It cannot be interpreted literally, or definitionally, in the context of the factory. If one did one might wander all over the factory looking for a box that was red, while passing by any number of "red boxes" in which he ought to toss the scrap piece of lumber.
So by this thinking it makes perfect sense to talk of the blue red box, meaning the scrap lumber box that for some reason or an other is actually blue.
Perhaps "blue jeans" a term that is to be interpreted definitionally or nondefinitionally. Does it make sense to talk of a pair of red blue jeans, in which red is the color and "blue jeans" means pants of a certain material and style?
What are the dimensions of a two by four? Is that an obvious question? Or to change it a little, what are the dimensions of an eight foot two by four? If one thinks definitionally the answer is that it is eight feet long, two inches thick and four inches wide. But if you went to a lumber yard and asked for an eight foot two by four that is not what you would be led to. If you wanted exactly that it would probably have to be custom cut. An eight foot two by four would be only about one and a half inches thick, three and a half inches wide, and perhaps eight feet long, or perhaps ninety-two and five eighths inches long. That's simply the way lumber is in America.
At times I wondered just what a "fell swoop" is. It seems obvious that "swoop" must be a noun, and "fell" must be an adjective. But just what does "fell" mean. How is a "fell swoop" different than a simple "swoop"? The point here is that we do not always think definitionally. Our language doesn't work that way, not always. So perhaps it should not be surprising if sometimes we fail to think definitionally at all times. A lot of language decoding depends on knowing and interpreting idioms. Thus we may fail to think definitionally when we should think definitionally.
So it is a possibility that Chelsea's difficulty is simply due to not thinking definitionally when she needs to think definitionally.
Following these examples we can see the possibility that a child reading "fifteenth part" in the sentence "each of the robbers expected to receive fifteenth share of the loot", might have no trouble at all interpreting "fifteenth" definitionally, one part out of fifteen. But it is also possible that it would be interpreted more generally, as somehow meaning something not too definite. Then in an arithmetic problem, such as "what is one divided by one fifteenth?" that definitional interpretation might not be called into play, or it might be called into play, but somehow subverted by some extraneous thinking due to the previous instruction. The meaning of "fifteenth" in this context might be "some number, expressed as a fraction". The pattern of thinking of previous problems may cause the child to expect that some process should be followed, and this thinking may make the child blind to the obvious. It may well be that explaining this problem to a child might be done, at least in part, by explicitly bringing forth consideration of that definition. But that would not necessarily be easy or obvious. It might be one of those situations in which the teacher struggles to explain and the child struggles to understand, and eventually they get there.
In the video the guy attempts to explain the problem by eliciting the definition of miles per hour. Is there any better way to do it? He says, in the video, that his is using math and science. I would disagree. He is using only language, common everyday language. "Miles", "per", and "hour" are all common vocabulary words. If you think definitionally then it is obvious that the answer is "one hour" to the problem in the video. Neither math nor science is involved, only language, knowing the meaning of the words and interpreting definitionally. The woman had trouble doing this. Therefore the question becomes, why did she have trouble doing it? The answer, or at least one possible answer, it seems to me, would be that she had have come to think of "miles per hour" in a nondefinitional way. "Forty miles per hour" in her mind means forty units of speed. "Mile per hour" is a unit of speed, nothing more, just as "two by four" is descriptive, but not definitive.
So all this is understandable. Where does it lead?
It does not lead, in my view, to any obvious way to explain to Chelsea that it takes an hour to go eighty miles if you are traveling at eighty miles per hour. The guy tries to invoke the meaning of miles per hour, and I don't know how to improve on that. One can imagine that at some point a light bulb goes on in Chelsea's head and it all makes sense. This reminds me of the many, many times when I would try to help students in my office with the math that I was trying to teach. Again and again I would be struggling to find the right words to explain something or other, and the student would suddenly say, "Oh, I get it . . ." Assembling a thought into a coherent string of words is important of course. That's what they guy in this video was trying hard to do, just like I would always try to do when helping students in my office But that is only part of the battle. Then the recipient of these words has to take the thoughts of the words, and all other relevant thoughts from whatever source, and assemble them together into a meaningful structure.
The constructivists have always been right. We have to construct meaning. Chelsea, we hope, will eventually take the thoughts the guy presented, and her own thoughts, sort it all out in her mind, and understand. Then it will seem very obvious to her. She will have constructed the understanding out of the materials she had available, but only after a laborious process of trying to fit the parts together.
The constructivists are right, of course, but it has always been a bit of a mystery to me how they can start with a correct premise, that each person has to construct meaning, and in that find support for the ideas of fuzzy math. But that is not our main concern at the moment.
I think this video shows how unrelated thoughts can sabotage our thinking. Chelsea was convinced that thoughts such as how many revolutions the tire makes in a mile, or how fast she can run, were relevant to the problem of the moment. Therefore she kept dealing with those thoughts. She felt if she could only assemble those thoughts in the right way they would lead to a conclusion, or so it appears to me at least. This complicates the problem of explaining the problem to her. It takes time to explain to her that those thoughts are irrelevant. Sometimes in such a situation you can say "forget all that, just concentrate on . . ." Other times each extraneous idea must be carefully considered and dismissed only after this careful thought warrants it. This can take a lot of time.
I have offered some ideas of how Chelsea's thinking might be on the problem, but I can't say anything definite. That is normally the case when I help students in my office on math. Sometimes a particular difficulty becomes apparent that can be corrected. More often we stumble around, just as Chelsea and her husband are doing in this clip. At the end of this stumbling often the student leaves with something worthwhile. Sometimes the student leaves with a clear cut understanding of whatever the problem was. But more commonly the result is not quite so happy. Often success consists of a better understanding of the math in question, or strategies to try, or resources to look into. This can sound a little discouraging, but my view is that that is the nature of learning something like math. Stumbling around is essential to constructing meaning. Constructing meaning, or constructing understanding, means taking the available knowledge and understanding and trying to assemble the parts in ways that make sense. The efficiency of this construction, this stumbling around, is important, but in most cases the efficiency is not crucial. What is really crucial is that the construction simply be done, that the learner gets the parts put together, whether it took a lot of time or not.
This clip makes it apparent to me that miles per hour problems ought to be taught somewhere in the teaching of arithmetic. Only doing problems will enable a person like Chelsea to understand them. In this clip, it appears to me at least, Chelsea is faced with a problem in isolation. Had she done a number of problems in the context of her arithmetic instruction I think it would be understandable now. She would construct the meaning with the help of a teacher and in the context of classroom instruction in which her classmates are doing the same thing. I can well imagine that it would be harder for Chelsea than for many of her classmates, but that is the case in any learning. It is harder for some than for others.
So when, and in what context, should the construction of meaning of the arithmetic of rates be done? Should it be done in the third grade, or the fifth, or can it wait until algebra? Or does it need to be done at all in the teaching of math for most students? As I have mentioned I remember no arithmetic problems in my own elementary education background. I can say that that lack didn't hurt me, but I'm a little better than average at that sort of thing. Chelsea, we might guess, is a little worse than average at that sort of thing. It seems to me that mostly likely Chelsea never had rate problems in fourth, fifth, sixth, seventh, or eighth grades, and the result is what we see in the video.
Such problems could come anytime after students get a reasonably firm grasp of multiplication. I assume that would perhaps be in fourth grade, and that's why I mentioned fourth, fifth, sixth, seventh, or eighth grades. If students can multiply 36 times 14, then problems such as, "A train travels west from Chicago for fourteen hours at the speed of thirty six miles per hour. How many miles does the train go in that time?" can make sense.
I can't say where in the math curriculum such problems should be done, but I feel they definitely should be somewhere. Not all students necessarily need them, but surely Chelsea does.
And when division is reasonably firm in a student's mind a problem in which total miles is given and either rate of speed or number of hours is asked for can make sense. A problem of this time would be something like, "How many hours does it take to go from Omaha to St. Louis, a distance of four hundred and thirty nine miles, if you travel at a constant speed of sixty-two miles per hour?" Again, many of us would have no trouble with this problem if it is never encountered in school, but Chelsea would.
And perhaps variations of such problems are appropriate and helpful, such as how many minutes does it take to go one mile at x miles per hour, or which is faster, nine miles per hour or a mile in seven minutes. And how about a question mentioned in the video, how many times does a tire turn in one mile?
All this takes time, of course, and time is a precious resource in teaching and learning. Chelsea probably spent something like 180 hours of class time in the fifth grade on math, plus more hours doing math homework outside of class. Was it time well spent? How could we know?
In watching Chelsea in this clip an unpleasant vision enters my mind. I see a young Chelsea, let's say eighth grade, in a math class. The class has broken up into groups and are supposed to collaborate on solving the problem, "A tire has a diameter of 26 inches. How many times will it turn in going one mile?" In this setting Chelsea does exactly what she is doing in this clip. And her group mates are doing much the same. In the clip she keeps coming back to her personal experiences, how long it takes her to run a mile, and so on. One can imagine this line of thinking coming from collaborative effort in school. Recalling personal experiences only marginally related is not a good way to approach a math problem.
In this imaginary scenario one might imagine another eighth grader who's got the problem figured out sitting resignedly while classmates stumble around with Chelsea and get nowhere. Eventually time is up, each committee reports it's accumulated nonsense to the class as a whole and life goes on. This scenario has appeal to some, but not to me. Constructivism makes sense, but somehow it has become a fad, and those who latch on to fads embrace the term while embracing the wasteful practices that make it very hard for a student to actually construct knowledge.
Did Chelsea's thinking in this video derive in part from years of fuzzy math in school? Or is my unpleasant vision that I just described purely a figment of my imagination unconnected to reality? I'll never know.